Twisted generalized Weyl algebras and primitive quotients of enveloping algebras. (English) Zbl 1352.16027
In this paper, the authors use multiquivers to construct new examples of twisted generalized Weyl algebras. The algebras in these examples carry a canonical representation by differential operators and are shown to be universal among all twisted generalized Weyl algebras having such a representation. The generalized Cartan matrix of the twisted generalized Weyl algebras turns out to be the same as for the Dynkin diagram associated to the multiquiver. Furthermore, it is shown that the constructed twisted generalized Weyl algebras contain graded homomorphic images of the universal enveloping algebra of the positive part of the corresponding Kac-Moody algebra. Finally, a connection to primitive quotients of the universal enveloping algebra of simple finite dimensional Lie algebras is established: it is shown that such a quotient is graded isomorphic to a twisted generalized Weyl algebra if and only if the corresponding primitive ideal is the annihilator of a completely pointed simple weight module.
Reviewer: Volodymyr Mazorchuk (Uppsala)
MSC:
| 16S30 | Universal enveloping algebras of Lie algebras |
| 16S99 | Associative rings and algebras arising under various constructions |
| 16W50 | Graded rings and modules (associative rings and algebras) |
| 16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |
| 17B35 | Universal enveloping (super)algebras |
Keywords:
generalized Weyl algebra; enveloping algebra; primitive ideal; quotient; differential operatorReferences:
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